Question

What should be added to 6x^5+ 4x^4– 27x^3– 7x^2 – 27x – 6 so that the resulting polynomial is exactly
divisible by (2x - 3).​

Answer 1

this is your answer dear plz mark as a brain list

$6x5+4x4−7x2−27x−6=(6x5−9x3)+(4x4−6x2)+(9x3−27x2)−(x2−32)−27x2−152=3x3(2x2–3)+2x2(2x2–3)+9x2(2x2–3)−12(2x2–3)−27x2−152=(2x2–3)(3x3+2x2+9x2−12)−(27x2+152)6x5+4x4−7x2−27x−6=(6x5−9x3)+(4x4−6x2)+(9x3−27x2)−(x2−32)−27x2−152=3x3(2x2–3)+2x2(2x2–3)+9x2(2x2–3)−12(2x2–3)−27x2−152=(2x2–3)(3x3+2x2+9x2−12)−(27x2+152)</p><p></p><p>From the above simplification of the given polynomial we can conclude that (27x2+152)(27x2+152) must be added to the first polynomial to make it divisible by (2x2–3)(2x2–3).</p><p></p><p>EDIT 1 :</p><p></p><p>To explain the method further I added a second intermediate line in the process. The trick is to write the third line before the second. How you get the third line? Well just consider the term with the highest degree (e.g. 6x56x5 here) of xx, and look for the co-efficient (e.g. 3x33x3 here) that need to be removed from it to get the highest degree term of the divisor (e.g. 2x22x2 here).Now multiply the co-efficient with the whole divisor. Don’t worry about the extra terms you get, we will equate them out later.Now the first thing you do is writing the multiplied part of the polynomial in the second line within a parenthesis.Next you move to the next highest degree term in the polynomial and work out the same process as above. You have to keep an eye for the polynomial building in the second line also, if you have a term there with higher degree of xx then you work them out before going back to the original polynomial.When you are left with lower degree terms than the divisor then you stop the iteration as this remaining portion will be your remainder.</p><p></p><p>P.S. : You can always use division by polynomials to find out the remainded \: </p><p></p><p></p><p></p><p></p><p>$